Monomial modular representations and construction of the Held group (Q1815024)

The author continues his study of symmetric presentations of interesting groups, by generalising to a symmetric set of generators, each of order bigger than 2, permuted monomially by a suitable `control group'. In the present paper, he defines natural monomial representations of the three groups \(\text{GL}_2(3)\), \(\text{SL}_2(5)\), and \(3\cdot A_7\), and the corresponding action on symmetric generators of order 3, 5 and 7 respectively. (For technical reasons, he also adjoins outer automorphisms of the two latter groups.) It is then shown how in each case a single extra relation, namely the simplest possible relation which does not imply collapse, yields a presentation of the Mathieu group \(M_{11}\), the unitary group \(3\cdot U_3(5):2\) and the Held group respectively. Of particular interest is the fact that five of the conjugacy classes of maximal subgroups of the Held group are represented very easily in terms of the standard generators, and four more are obtained with only a little more difficulty.